Les graphes 2-reconstructibles indécomposables

Boussaïri, Abderrahim; Chaïchaâ, Abdelhak

doi:10.1016/j.crma.2007.05.028

Cited by 3 publications

(1 citation statement)

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“…Properties of the prime substructures of a given prime structure were determined by Schmerl and Trotter Schmerl and Trotter (1993) in their fundamental paper. Indeed, several papers within the same sphere of reference have then appeared ; ; Bouchaala et al (2013); Boudabbous (2016); Boussaïri and Chaïchaâ (2007); Ehrenfeucht et al (1997); Ille (1997); Elayech et al (2015); Pouzet and Zaguia (2009); Sayar (2011)). For instance, the path defined on N n = {1, ..., n}, denoted by P n , is prime for n ≥ 4.…”

Section: Introduction and Presentation Of The Resultsmentioning

confidence: 99%

(-k)-critical trees and k-minimal trees

Marweni

2021

Preprint

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In a graph G = (V, E), a module is a vertex subset M of V such that every vertex outside M is adjacent to all or none of M . For example, ∅, {x} (x ∈ V ) and V are modules of G, called trivial modules. A graph, all the modules of which are trivial, is prime; otherwise, it is decomposable.there is some k-vertex set X of vertices such that there is no proper induced subgraph of G containing X is prime. From this perspective, I. Boudabbous proposes to find the (−k)-critical graphs and k-minimal graphs for some integer k even in a particular case of graphs. This research paper attempts to answer I. Boudabbous's question. First, it describes the (−k)-critical tree. As a corollary, we determine the number of nonisomorphic (−k)-critical tree with n vertices where k ∈ {1, 2, n 2 }. Second, it provide a complete characterization of the k-minimal tree. As a corollary, we determine the number of nonisomorphic k-minimal tree with n vertices where k ≤ 3.

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Section: Introduction and Presentation Of The Resultsmentioning

confidence: 99%

(-k)-critical trees and k-minimal trees

Marweni

2021

Preprint

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Description of the Tournaments Which are Reconstructible from Their k-cycle Partial Digraphs for $$k \in \{3, 4\}$$ k ∈ { 3 , 4 }

Hamadou

Boudabbous

Amri

et al. 2016

Graphs and Combinatorics

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International audienceIf T is a tournament on n vertices and k is an integer with , then the k-cycle partial digraph of T, denoted by T(k), is the spanning subdigraph of T for which the arcs are those of the k-cycles of T. In 1989, Thomassen proved that given two irreducible tournaments T and on the same vertex set with at least 6 vertices, if , then . This result allows us to introduce the following notion of reconstruction. A tournament T is reconstructible from its k-cycle partial digraph whenever is isomorphic to T for each tournament such that is isomorphic to T(k). In this paper, we give a complete description of the tournaments which are reconstructible from their k-cycle partial digraphs where . For , our proof is based on the above result of Thomassen. For , we introduce and study the tournaments for which all modules are irreducible. We use properties of the dilatation operator to describe such tournaments, and then we obtain a new modular decomposition of tournaments sometimes finer than that of Gallai

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Primality, criticality and minimality problems in trees

Marweni

2022

Discrete Math. Algorithm. Appl.

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In a graph [Formula: see text], a module is a vertex subset [Formula: see text] of [Formula: see text] such that every vertex outside [Formula: see text] is adjacent to all or none of [Formula: see text]. For example, [Formula: see text], [Formula: see text][Formula: see text] and [Formula: see text] are modules of [Formula: see text], called trivial modules. A graph, all the modules of which are trivial, is prime; otherwise, it is decomposable. A vertex [Formula: see text] of a prime graph [Formula: see text] is critical if [Formula: see text] is decomposable. Moreover, a prime graph with [Formula: see text] noncritical vertices is called [Formula: see text]-critical graph. A prime graph [Formula: see text] is [Formula: see text]-minimal if there is some [Formula: see text]-vertex set [Formula: see text] of vertices such that there is no proper induced subgraph of [Formula: see text] containing [Formula: see text] is prime. From this perspective, Boudabbous proposes to find the [Formula: see text]-critical graphs and [Formula: see text]-minimal graphs for some integer [Formula: see text] even in a particular case of graphs. This research paper attempts to answer Boudabbous’s question. First, we describe the [Formula: see text]-critical tree. As a corollary, we determine the number of nonisomorphic [Formula: see text]-critical tree with [Formula: see text] vertices where [Formula: see text]. Second, we provide a complete characterization of the [Formula: see text]-minimal tree. As a corollary, we determine the number of nonisomorphic [Formula: see text]-minimal tree with [Formula: see text] vertices where [Formula: see text].

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Les graphes 2-reconstructibles indécomposables

Cited by 3 publications

References 5 publications

(-k)-critical trees and k-minimal trees

(-k)-critical trees and k-minimal trees

Description of the Tournaments Which are Reconstructible from Their k-cycle Partial Digraphs for $$k \in \{3, 4\}$$ k ∈ { 3 , 4 }

Primality, criticality and minimality problems in trees

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